The (pre-)Fredholm modules are the building blocks of the Kasparov
\(K\!K\) -theory (see [1]} or [2]} for more details). In a nutshell, we can say that the
Kasparov \(K\!K\) -theory is a homology theory in which the cycles are
(pre-)Fredholm modules. Elements of the \(K\!K\) -homology group are equivalence
classes of (pre-)Fredholm modules modulo homotopy. With this in mind, we can
consider the triplets \((\rho ,{H}_4,F_{B,\varepsilon })\)  as cycles of the \(K\!K\) -homology
\(K\!K({C}_B,\) . Since the map \((0,+\infty )\ni \varepsilon \mapsto F_{B,\varepsilon }\)  is norm-continuous it
follows that all these cycles define a unique class of \(K\!K({C}_B,\)  that
will be denoted with \([F_B]\) . Moreover, since \(\lim \limits _{\varepsilon \rightarrow 0^+}F_{B,\varepsilon }=F_{B,0}\) 
when restricted to \(V_0^\bot \) , and the remainder in the orthogonal complement is a
compact perturbation (in the sense of [2]}) it follows that the class \([F_B]\)  contains also the \(\varepsilon =0\) 
regularization described in Remark REF .
